Update documentation sites base branch (#13592)

This PR updates scripts and documentation related to SDK releases which
deploy API reference docs to 3 documentation sites sites:
- `android-docs` - Android maps, navigation and search SDKs
- `ios-sdk` - iOS Maps, Navigation and seearch SDKs
- `api-documentation` - Web apis

## Summary
The Docs team stopped using `MBX Publisher` for deployments. Publisher
used a prefixed `publisher-` on deployment branches. As a housekeeping
measure @mapbox/docs removed the `publisher-` prefix from base branches
of all docs sites. These scripts reference that base branch for checks
and deployments. The corresponding files have been updated to reference
the new base branch. (`production`)

---------

Co-authored-by: Changelog autocreator bot <github-actions[bot]@users.noreply.github.com>
(cherry picked from commit 76f3250f1e9eb1fa73a51754c30cf2f67787fae9)
GitOrigin-RevId: f788b126ad348f47ac2f234c9fa38fd3d27289ba

Author: AndrewSepic

CHANGELOG.md

@@ -6,6 +6,10 @@
 
 * Fixed a possible hang during `MapboxSpeechSynthesizer` initialization.
 
+### Other changes
+
+* Updated API reference docs deploy script to target `production` branch instead of `publisher-production`.
+
 ## 3.24.0-rc.1
 
 ### Packaging